Average-Case Complexity of the Min-Sum Matrix Product Problem

Abstract

We study the average-case complexity of min-sum product of matrices, which is a fundamental operation that has many applications in computer science. We focus on optimizing the number of “algebraic” operations (i.e., operations involving real numbers) used in the computation, since such operations are usually expensive in various environments. We present an algorithm that can compute the min-sum product of two $n \times n$ real matrices using only $O(n^2)$ algebraic operations, given that the matrix elements are drawn independently and identically from some fixed probability distribution satisfying several constraints. This improves the previously best known upper-bound of $O(n^2\log n)$. The class of probability distributions under which our algorithm works include many important and commonly used distributions, such as uniform distributions, exponential distributions, and folded normal distributions.

In order to evaluate the performance of the proposed algorithm, we performed experiments to compare the running time of the proposed algorithm with algorithms in [7]. The experimental results demonstrate that our algorithm achieves significant performance improvement over the previous algorithms.

This work was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [Project No. CityU 122512].